Question: Simplify; express your answer in exponential form. Assume $k\neq 0, t\neq 0$. $\dfrac{{(k^{4}t^{-3})^{-4}}}{{(kt)^{-5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{4}t^{-3})^{-4} = (k^{4})^{-4}(t^{-3})^{-4}}$ On the left, we have ${k^{4}}$ to the exponent ${-4}$ . Now ${4 \times -4 = -16}$ , so ${(k^{4})^{-4} = k^{-16}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{4}t^{-3})^{-4}}}{{(kt)^{-5}}} = \dfrac{{k^{-16}t^{12}}}{{k^{-5}t^{-5}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-16}t^{12}}}{{k^{-5}t^{-5}}} = \dfrac{{k^{-16}}}{{k^{-5}}} \cdot \dfrac{{t^{12}}}{{t^{-5}}} = k^{{-16} - {(-5)}} \cdot t^{{12} - {(-5)}} = k^{-11}t^{17}$